As mentioned above, \(q\) -TAZRP and \(q\) -TASEP have the stochastic duality. There is a natural particle-spacing duality between zero range process and simple exclusion process (SEP) models generally [1]}, such that if particles \(\lbrace  x_i(t) \rbrace \)  are in a simple exclusion process, then their spacings \(\lbrace  y_i = x_i - x_{i + 1} \rbrace \)  are in a corresponding zero range process. While in [2]}, a different stochastic duality that is associated to a duality function is observed, and it is applied there to derive moment formulas of \(q\) -TASEP. Therefore it is natural that the exact formulas enjoyed by \(q\) -TASEP have a dual version for \(q\) -TAZRP, and vice versa, although the relation between the formulas may not be direct. However, there is a certain discrepancy between the results for \(q\) -TASEP and those for \(q\) -TAZRP. The \(q\) -TASEP as a specialization of Macdonald process naturally contains parameters, which are independent of \(q\) , for the jumping rates of particles, so the particles may not be identical, but some are fast particles and some are slow. In the study of \(q\) -TAZRP, Korhonen and Lee assume that the jump rate of a particle only depends on the height of the stack where the particle belongs, and the model has no parameter other than \(q\) .
